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符號辭彙表

符號

含義

i

-1的平方根The square root of minus one

f(x)

自變數為x的函數f的值The value of the function f at argument x

sin(x)

自變數為x的正弦函數值

The value of the sine function at argument x

exp(x)

自變數為x的指數函數值，也記作ex

The value of the exponential function at argument x. This is often written as ex

a^x

a的x次方；有理數x定義為反函數

The number a raised to the power x; for rational x is defined by inverse functions

ln x

exp x的反函數The inverse function to exp x

ax

與a^x相同Same as a^x

logba

為了得出a ，b; blogba = a

The power you must raise b to in order to get a; blogba = a

cos x

自變數為x的餘弦函數的值（正弦的餘）

The value of the cosine function (complement of the sine) at argument x

tan x

由 sin x/cos x得出Works out to be sin x/cos x

cot x

正切函數的餘的值或

cos x/sin x The value of the complement of the tangent function or cos x/sin x

sec x

正割函數的值，即

1/cos x Value of the secant function, which turns out to be 1/cos x

csc x

正割函數餘的值，即

1/sin x Value of the complement of the secant, called the cosecant. It is 1/sin x

asin x

自變數為x的正弦的反函數的值. 即

x = sin y The value, y, of the inverse function to the sine at argument x. Means x = sin y

acos x

自變數為x的餘弦的反函數的值. 即

x = cos y The value, y, of the inverse function to cosine at argument x. Means x = cos y

atan x

自變數為x的正切的反函數的值. 即 

x = tan y The value, y, of the inverse function to tangent at argument x. Means x = tan y

acot x

自變數為x的餘切反函數的值. 即 

x = cot y The value, y, of the inverse function to cotangent at argument x. Means x = cot y

asec x

自變數為x的正割的反函數的值. 即 

x = sec y The value, y, of the inverse function to secant at argument x. Means x = sec y

acsc x

自變數為x的反割的反函數的值. 即 

x = csc y The value, y, of the inverse function to cosecant at argument x. Means x = csc y

θ

角的標準符號. 除注明外一般用弧度制，特別用於atan x/y 當 x, y, z 是描述三維空間裏的點的時候

A standard symbol for angle. Measured in radians unless stated otherwise.

Used especially for atan x/y when x, y, and z are variables used to describe point in three dimensional space

i, j, k

分別是x 、y、 z方向上的單位向量

Unit vectors in the x y and z directions respectively

(a, b, c)

x 分量為a，y分量為 b，z分量為c的向量

A vector with x component a, y component b and z component c

(a, b)

x 分量為a，y分量為 b的向量

A vector with x component a, y component b

(a, b)

向量a和b的內積The dot product of vectors a and b

a•b

向量a和b的內積The dot product of vectors a and b

(a•b)

向量a和b的內積The dot product of vectors a and b

|v|

向量v的量The magnitude of the vector v

|x|

X的絕對值The absolute value of the number x

Σ

表示和，它的參數和起始標號常標示在此符號的下方而終止標號則標示於其上方。

例如從j=1到n的和可寫為 ， 意為1 + 2 + … + n Used to denote a summation,

usually the index and often their end values are written under it with upper end value above it.

For example the sum of j for j=1 to n is written as . This signifies 1 + 2 + … + n

M

用於表示數值或其他物件的矩陣或陣列。

Used to represent a matrix or array of numbers or other entities

|v>

行向量，其物件排列成行，可視為k乘1的矩陣。

A column vector, that is one whose components are written as a column and treated as a k by 1 matrix

<v|

列向量，或1乘k的矩陣。

A vector written as a row, or 1 by k matrix

dx

變數x的無窮小的變化; dy, dz, dr 等同理

An "infinitesimal" or very small change in the variable x; also similarly dy, dz, dr etc...

ds

非常小的距離變化A small change in distance

ρ

變數(x2 + y2 + z2)1/2 或球體座標系中到原點的距離

The variable (x2 + y2 + z2)1/2 or distance to the origin in spherical coordinates

r

變數(x2 + y2)1/2或在三維座標或極座標中到Z軸的距離

The variable (x2 + y2)1/2 or distance to the z axis in three dimensions or in polar coordinates

|M|

矩陣M的行列式（ 由他的行或列確定面積或側面平行的區域的體積）

The determinant of a matrix M (whose magnitude is the area or volume of the parallel sided region determined by its columns or rows)

||M||

矩陣M的行列式的量，是體積，面積或超體積

The magnitude of the determinant of the matrix M, which is a volume or area or hypervolume

det M

M的行列式The determinant of M

M-1

矩陣M的逆The inverse of the matrix M

v×w

向量v和向量w的內積或叉積。

The vector product or cross product of two vectors, v and w

θvw

向量v和w的夾角。The angle made by vectors v and w

A•B×C

標量三重積，或是由列A,B,C所構成矩陣的行列式值。

The scalar triple product, the determinant of the matrix formed by columns A, B, C

uw

向量w方向上的單位向量，它的含義與w/|w|相同

A unit vector in the direction of the vector w; it means the same as w/|w|

df

函數f的無窮小變化，小至對於所有函數皆可利用線性近似表示。

A very small change in the function f, sufficiently small that the linear approximation to

all relevant functions holds for such changes

df/dx

f對x的微分，是對於函數f的線性逼近。

The derivative of f with respect to x, which is the slope of the linear approximation to f

f '

f對變數的微分，通常對x。

The derivative of f with respect to the relevant variable, usually x

∂f/∂x

函數f對x的偏微分，固定y和z。一般來說函數f對某一指定變數q的偏微分

所代表的是在其餘變數固定的情形之下，df對dq的比值。

由於對於哪個變數保持固定可能產生誤解，所以應該明確標注出來。

The partial derivative of f with respect to x, keeping y, and z fixed.

In general a partial derivative of f with respect to a variable q is the ratio of df to dq when certain other variables are held fixed.

Where there is possible misunderstanding over which variables are to be fixed that information should be made explicit

(∂f/∂x)|r,z

固定r和z，函數f對x的偏微分。

The partial derivative of f with respect to x keeping r and z fixed

grad f

各分量分別為f對x,y和z的偏微分的向量場：

[(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k；稱作f的梯度

The vector field whose components are the partial derivatives of the function f with respect to x, y and z: [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k; called the gradient of f

∇

向量運算(∂/∂x)i + (∂/∂x)j + (∂/∂x)k, 讀作 "del"

The vector operator (∂/∂x)i + (∂/∂x)j + (∂/∂x)k, called "del"

∇f

f的梯度；它與uw的內積為f在w方向上的方向導數

The gradient of f; its dot product with uw is the directional derivative of f in the direction of w

∇•w

向量場w的散度；為向量算符∇ 與向量w的內積，

或(∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z)  

The divergence of the vector field w; it is the dot product of the vector operator ∇ with the vector w, or (∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z)

curl w

向量算符∇與向量w的叉積。

The cross product of the vector operator ∇ with the vector w

∇×w

w的旋度，分量為[(∂fz /∂y) - (∂fy /∂z), (∂fx /∂z) - (∂fz /∂x), (∂fy /∂x) - (∂fx /∂y)]

The curl of w, with components [(∂fz /∂y) - (∂fy /∂z), (∂fx /∂z) - (∂fz /∂x), (∂fy /∂x) - (∂fx /∂y)]

∇•∇

拉普拉斯運算元，為微分運算元：(∂2/∂x2) + (∂/∂y2) + (∂/∂z2)

The Laplacian, the differential operator: (∂2/∂x2) + (∂/∂y2) + (∂/∂z2)

f "(x)

f對x的二階導數；亦為f '(x)的導數

The second derivative of f with respect to x; the derivative of f '(x)

d2f/dx2

f對x的二階導數The second derivative of f with respect to x

f(2)(x)

f對x的二階導數的另一種形式

Still another form for the second derivative of f with respect to x

f(k)(x)

f對x的k階導數；亦為f(k-1) (x)的導數

The k-th derivative of f with respect to x; the derivative of f(k-1) (x)

T

沿著曲線的單位切向量；若曲線為r(t)，

T = (dr/dt)/|dr/dt| Unit tangent vector along a curve; if curve is described by r(t), T = (dr/dt)/|dr/dt|

ds

沿著曲線的長度的微分A differential of distance along a curve

κ

曲線的曲率；為單位切向量對距離取導數所得向量的大小：

|dT/ds| The curvature of a curve; the magnitude of the derivative of its unit tangent vector with respect to distance on the curve: |dT/ds|

N

沿著dT/ds方向上的單位向量，與T垂直。

A unit vector in the direction of the projection of dT/ds normal to T

B

垂直與平面T 和 N的單位向量, 即面的曲率

A unit vector normal to the plane of T and N, which is the plane of curvature

τ

曲線的擾率; |dB/ds| The torsion of a curve; |dB/ds|

g

引力常數The gravitational constant

F

力學中力的標準符號The standard symbol for force in mechanics

k

彈簧的彈性係數The spring constant of a spring

pi

第I個質點的動量The momentum of the i-th particle

H

物理系統的哈密頓量，它的能量用{ri} 和 {pi}即 位置和動量表示

The Hamiltonian of a physical system, which is its energy expressed in terms of {ri} and {pi}, position and momentum

{Q, H}

Q 和 H的泊松括弧The Poisson bracket of Q and H

f(x)的反導數， 用x的函數表示出來An antiderivative of f(x) expressed as a function of x

f從a到b的定積分。當 f 為正值且 a < b , 那它代表 直線 y = a, y = b與x軸和函數曲線所圍成的面積

The definite integral of f from a to b. When f is positive and a < b holds, then this is the area between the x axis the lines y = a, y = b and the curve that represents the function f between these lines

L(d)

在等區間大小d下，f在每一區間中取左端點所得的黎曼和。

A Riemann sum with uniform interval size d and f evaluated at the left end of each subinterval

R(d)

在等區間大小d下，f在每一區間中取右端點所得的黎曼和。

A Riemann sum with uniform interval size d and f evaluated at the right end of each subinterval

M(d)

在等區間大小d下，f在每一區間中取極大值點所對應的點所得的黎曼和。

A Riemann sum with uniform interval size d and f evaluated at the maximum point of f in each subinterval

m(d)

在等區間大小d下，f在每一區間中取極小值點所對應的點所得的黎曼和。

A Riemann sum with uniform interval size d and f evaluated at the minimum point of f in each subinterval